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  "body": "\n## Definition\n\nLet $A$ and $B$ be distinct points of the Euclidean space $\\mathbb{R}^n$, and let $M = \\frac{1}{2}(A + B)$ be their [[midpoint|0S706JBZBG]]. Let $\\cdot$ be the dot product.\nThe perpendicular bisector of $A$ and $B$ is the set\n$$\\{\\, X \\in \\mathbb{R}^n \\;:\\; (X - M) \\cdot (B - A) = 0 \\,\\}.$$\n\n## Remarks\n\nA point $X$ is on the perpendicular bisector if and only if $d(X, A) = d(X, B)$, where $d$ is the [[Euclidean distance|0NPPGHQNXQ]].\nIt is a hyperplane with the [[normal vector|0WQJJCCTC8]] $B - A$. In the plane it is a line, and in $\\mathbb{R}^3$ it is a plane.\nMathlib's `AffineSubspace.perpBisector` also allows $A = B$. Then the set is the whole space.\nThe perpendicular bisectors of the three sides of a triangle in the plane meet at the [[circumcentre|0F98FZK14A]].\n\n## Explanations\n\n### L1 human\n\nThe perpendicular bisector of a segment cuts it in half at a right angle. Each point on it is as far from one end of the segment as from the other. For the points $(0, 0)$ and $(2, 0)$, the perpendicular bisector is the vertical line $x = 1$.\n",
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